In complex geological environments, wave equation migration is recognized to be the best imaging technique currently available for imaging seismic data. Wave equation migration comes in two forms usually called WEM and RTM. In WEM (“Wave Equation Migration”) energy is back propagated from the receivers using a one-way wave equation, and forward propagated from the corresponding source. The wave fields are cross correlated at each image point to create the subsurface seismic image. This method can produce good images for reflectors whose dip is relatively shallow. In RTM (“Reverse Time Migration”) the wave field at the receivers is back-propagated using a two-way wave equation, and is cross correlated with energy forward propagated from the source. This method can produce good images at all reflector dips, but is more expensive than WEM by a factor typically in the range of 4-10. However it is not straightforward with either method to efficiently produce common reflection angle gathers. Such gathers are useful in interpretation of the seismic images and also in velocity analysis. It is also possible to work with surface offset gathers; however these are less useful than angle gathers in complex imaging situations because they do not handle multipathing.
Current Technology
One way of deriving angle domain image gathers (Xie and Wu, 2002) uses local plane wave decomposition. This method has the disadvantage that it requires computation of a local Fourier transform, and is therefore not computationally efficient if angle gathers are required at many image points.
In wave equation migration methods generally, the image is produced by an imaging condition such as:DM({right arrow over (x)})=∫dωps({right arrow over (x)},ω)pr*({right arrow over (x)},ω)  (1)where the subscripts s and r respectively label the source and receiver side wave fields, the source side wave field being forward propagated from a source location, and the receiver side wave field being back propagated from receiver locations. As is well known all such cross correlations may be performed in either the frequency or the time domain. For the sake of brevity, in this document the equations are written in the frequency domain, but should be understood to apply in either domain. The symbol * means the complex conjugate. The label M refers to the fact that the data have been migrated to form an image at point {right arrow over (x)}. [Notation: in the following text, all vectors are presumed to be in 3D and are denoted by symbols with an arrow over them (e.g. {right arrow over (x)}). Symbols with a caret over them (e.g. {circumflex over (n)}) are unit vectors.]Equation 1 refers to the simplest type of model which only includes P-waves in an isotropic medium. The general case will be discussed later in connection with equation 12.
Another way of creating angle gathers (Sava and Fomel, 2005) displaces image points from the source and receiver side wave fields ps and pr, producing an image DM by cross correlating as follows:DM({right arrow over (x)},{right arrow over (h)})=∫dωps({right arrow over (x)}−{right arrow over (h)},ω)pr*({right arrow over (x)}+{right arrow over (h)},ω)  (2)This cross correlation step in processing is a generalization of the previous imaging condition and would normally replace that imaging condition in wave equation based imaging. In this case, the output is subsurface offset gathers labeled by the parameter {right arrow over (h)}. This is a non-local method that may smear the spatial resolution of the output. A further problem with this approach is that it requires the computation and storage of data volumes for each value of {right arrow over (h)}. This approach may leads to impractical quantities of data, especially in 3D unless the 3D image space {right arrow over (x)} is sampled on a coarse grid.